18 LOCALLY COMPACT SPACES 65

3. Let E be a compact ultrametric space (Section 3.8, Problem 4), d the distance on E.
Show that for every *0 •= E, the image of E by the mapping x -»• d(x0 , x) is an at most
denumerable subset of the interval [0, + QO [ in which every point (with the possible
exception of 0) is isolated (3.10.10). (For any r = </(r0 , *) > 0, consider the l.u.b. of
^(*o , y) on the set of points y such that d(xQ ,y)<rt and the g.l.b. of d(x0 , z) on the
set of points z such that d(xQ , z) > r; use Section 3.8, Problem 4).

4. Let E be a compact metric space, d the distance on E,/a mapping of E into E such that,
for any pair (x, y) of points of E, d(f(x),f(y)) ^ d(x, y). Show that /is an isometry ofE
onto E. (Let a, b be any two points of E; put /„ =/M,i o/f an =/„(«), bn=fn(b); show
that for any e > 0 there exists an index k such that d(a, ak) ^ s and d(b, bk) ^ e (consider
a cluster value of the sequence (#„)), and conclude that /(E) is dense in E and that

5. Let E, E' be two metric spaces, /a mapping of E into E'. Show that if the restriction of/
to any compact subspace of E is continuous, then /is continuous in E (use (3.13.14)).

6. Let E, E' be two metric spaces, / a continuous mapping of E into E', K a compact
subset of E. Suppose the restriction / 1 K of /is mjective and that for every ;ceK,
there is a neighborhood V* of x in E such that the restriction / 1 V* of /is injective.
Show that there exists a neighborhood U of K in E such that the restriction / 1 U is
injective (use contradiction and (3.17.11)).

18. LOCALLY COMPACT SPACES

A metric space E is said to be locally compact if for every point x e E,
there exists a compact neighborhood of x in E. Any discrete space is locally
compact, but not compact unless it is finite (3.16.3).

(3,18.1) The real line R is locally compact but not compact.

This follows immediately from the Borel-Lebesgue theorem (3.17.6).

(3.18.2) Let A be a compact set in a locally compact metric space E. Then
there exists an r > 0 such that Vr (A) (see Section 3.6) is relatively compact
inE.

For each x e A, there is a compact neighborhood V^ of x; the ^x form
an open covering of A, hence there is a finite subset {xl9..., xn} in A such that

n

the tfx (I ^ i^ n) form an open covering of A. The set U = (J VJC| is com-

i=l

pact by (3.17.8) and is a neighborhood of A; hence the result, by applying
(3.17.11).d)).